Let $X$ be a smooth complete complex (algebraic) 3-fold, $D$ an effective divisor on $X$, and $C$ a smooth integral curve contained in the support of $D$. Let $X'$ be the blowup of $X$ along $C$, and denote by $D'$ the strict transform of $D$, $E$ the exceptional divisor, and $C'=D' \cap E$ the (set-theoretic) intersection.

Then is it true that $C \cong C'$, $D' \cong D$, and the normal bundles $N_{C/D} \cong N_{C'/D'}$ under the identification of $C \cong C'$ ? Are the degrees of the normal bundles the same?